Jamie Snape

Applications of Elliptic Functions in Classical and Algebraic Geometry

Abstract

The theory of elliptic functions offers abundant applications in arithmetic, dynamics, and even probability and statistics. Yet often overlooked is the significance of these functions within classical and algebraic geometry. This dissertation illustrates uses of the Jacobi and Weierstrass functions, taking motivation from the geometric path drawn by a swinging pendulum. Applications develop from the classical works of Gauss, Euler, and Fagnano on the rectification of the lemniscate and ellipse, through Poncelet's poristic polygons and Seiffert's spherical spiral, to modern day elliptic curve cryptography inspired by the work of Abel. The lesser-known nine circles theorem forms the conclusion, demonstrating the power of elliptic functions in simplifying complex proofs of elementary geometry.

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Citation

Jamie Snape, Applications of Elliptic Functions in Classical and Algebraic Geometry, Master's dissertation, Department of Mathematical Sciences, University of Durham, Durham, UK, 2004, advised by J. Vernon Armitage